Diagnosis
The Burst Sonic toothbrush is not designed for user-replaceable batteries. The handle is sealed shut to be waterproof.
How to Fix It
Officially, when the battery no longer holds a sufficient charge (you'll notice run time dropping from weeks to just days or it feeling weak), the standard fix is to replace the entire handle. Burst's business model is based on selling brush head subscriptions, and they often offer significant discounts or replacements for handles with dead batteries, especially if you're a subscriber. Your first step should be to contact their customer support.
If That Didn't Work
If you're out of warranty and handy, it is *physically possible* for a technician to pry open the sealed shell and replace the small lithium-ion cell inside. However, this is not recommended. You'll almost certainly break the waterproof seal, void any remaining warranty, and it's tough to find the exact battery specification. The cost and risk usually outweigh just buying a new handle on sale.
Heads Up
To maximize battery lifespan, follow the charge cycle: run it down every now and then before recharging, and don't leave it permanently on the charger. Treat it like your phone battery.